How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling

JinHyeong Park; Kyeong-Hwa Lee

doi:10.12973/eurasia.2016.1289a

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JinHyeong Park ¹, Kyeong-Hwa Lee ²

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¹ Myongji University² Seoul National University

Abstract

The purpose of this study is to design a modeling task to facilitate students’ inquiries into the chain rule in calculus and to analyze the results after implementation of the task. Researchers have pointed out that students may encounter difficulties in understanding the chain rule due to the formal approaches used to teach and learn this concept. In this study, we take a modeling approach to the teaching and learning of the chain rule by facilitating the generalization of students’ models and modeling activities. We assumed abductive reasoning to be one of the key factors which can support the generalization of students’ models and modeling activities. We believe that analogical reasoning and diagrammatic reasoning are key factors in fostering students’ use of abduction. As a result, we determined that the students’ models and modeling activities were generalized to the chain rule by their use of abductive reasoning, and the students found the chain rule to be a generalized rule for describing changes of various quantities.

Keywords

mathematical modeling
analogy
abduction
diagrammatic reasoning
chain rule

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2016, Volume 12, Issue 9, 2331-2352

https://doi.org/10.12973/eurasia.2016.1289a

Publication date: 29 Jun 2016

Article Views: 1566

Article Downloads: 1304

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References

Ärlebäck, J. B., Doerr, H., & O’Neil, A. H.(2013). A modeling perspective on interpreting rates of change in context, Mathematical thinking and learning, 15(4), 314-336.
Bailer-Jones, D. M. (1999), Tracing the Development of Models in the Philosophy of Science. In L. Magnani, N. J. Nersessian, & P. Thagard (Eds), Model-based reasoning in scientific discovery (pp. 23–40). Springer: NewYork.
Blum, W. et al. (2002). ICMI study 14: Applications and modelilng in mathematics education - Discussion document. Educational studies in mathematics, 51, 149-171.
Blum, W., & Borromeo Ferri, R. (2009). Mathematical Modelling: Can It Be Taught And Learnt? Journal of Mathematical Modelling and Application, 1(1), 45-58.
Carraher, D. W., Martinez, M. V. & Schliemann, A. D. (2008). Early algebra and mathematical generalization, ZDM, 40, 3-22
Carnap, R. (1939). Foundations of logic and mathematics, International Encyclopedia of Unified Science, Chicago University Press: Chicago.
Casti, J. (1989) Newton, Aristotle, and the Modeling of Living Systems. In Casti, J. & Anders, K. (Eds) Newton and Aristotle Towards a Theory of Models for Living System (pp. 47–98). Birkhuiser: Boston.
Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., John, D. S., Tolias, G. & Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? Journal of mathematical behavior, 16(4), 345-364.
Clement, J. J. (2009). Creative Model Construction in Scientists and Students: The Role of Imagery, Analogy, and Mental Simulation, Springer: New York.
Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analysis of classroom video recordings and transcripts. Educational Studies in Mathematics, 30(3), 213–228.
Cottrill, J. (1999). Students’ understanding of the concept ofchain rule in first year calculus and the relation to their understanding of composition of functions. Unpublished doctoral dissertation, Purdue University, Normal.
Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P. & Reed, H. (2012). Tool use and the development of the function concept: From repeated calculations to functional thinking, International Journal of Science and Mathematics Education, 10, 1243-1267.
Eco, U. (1983). Horns, hooves, insteps: Some hypotheses on three types of abduction. In U. Eco & T. Sebeok (Eds.), The sign of three: Dupin, Holmes, Peirce (pp. 198–220). Bloomington, IN: Indiana University Press.
Frejd, P. & Bergsten, C. (2016). Mathematical modelling as a professional task, Educational studies in mathematics, 91, 11-35.
Gordon, S. P. (2005). Discovering the chain rule graphically. Mathematics and Computer Education. 39(3), 195-197.
Gravemeijer, K. & Terwel, J. (2000) Hans Freudenthal: A mathematician on didactics and curriculum theory, Journal of Curriculum Studies, 32(6), 777-796.
Guicciardini, N. (2003). Newton's method and Leibniz's calculus, In H. N. Jahnke (Eds.) A History of Analysis (pp. 73-104). Providence, RI: American Mathematical Society.
Haciomeroglu, E. S. (2015). The role of cognitive ability and preferred mode of processing in students’ calculus performance, Eurasia Journal of Mathematics, Science & Technology Education, 11(5), 1165-1179.
Hashemi, N., Abu, M. S., Kashefi, H., Mokhtar, M. & Rahimi, K. (2015). Designing learning strategy to improve undergraduate students’ problem solving in derivatives and integrals: A conceptual framework, Eurasia Journal of Mathematics, Science & Technology Education, 11(2), 227-238.
Hitt, F. & González-Martín, A. S. (2015). Covariation between variables in a modelling process: The ACODESA (collaborative learning, scientific debate and self-reflection) method. Educational studies in mathematics, 88, 201-219.
Hoffmann, M. H. G. (2004). How to get it. Diagrammatic reasoning as a tool of knowledge development and its pragmatic dimension. Foundation of science, 9, 285-305.
Hoffmann, M. H. G. (2005). Signs as means for discoveries: Peirce and His Concepts of ''Diagrammatic Reasoning,' ''Theorematic Deduction," ''Hypostatic Abstraction," and "Theoric Transformation”, In M. H. G. Hoffinann, J. Lenhard & F. Seeger (Eds.) Activity and Sign - Grounding Mathematics Education (pp. 45-56). New York: Springer.
Lee, C.-Y., Chen, M.-J. & Chang, W.-L. (2014). Effects of the multiple solutions and question prompts on generalization and justification for non-routine mathematical problem solving in a computer game context, Eurasia Journal of Mathematics, Science & Technology Education, 10(2), 89-99.
Lee, K.-H. (2011). Modelling of and conjecturing on a soccer ball in a Korean eighth grade mathematics classroom, International Journal of Science and Mathematics Education, 9, 751-769.
Lee, K.-H. & Sriraman, B. (2011). Conjecturing via reconceived classical analogy, Educational studies in mathematics, 76, 123-140.
Lesh, R., & Doerr, H. (2003). In what ways does a models and modeling perspective move beyond constructivism? In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 519-556). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Lesh, R. & Harel, G.(2003). Problem solving, modeling, and local conceptual development, Mathematical thinking and learning, 5(2&3), 157-189.
Lesh, R. & Sriraman, B. (2005). Mathematics education as a design science, ZDM, 37(6), 490- 505.
Lutzer, C. V.(2003). Using motion to teach chain rule and u-substitution, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 13(1), 47-54.
Maaß, K. (2006). What are modelling competencies? ZDM, 38(2), 113-142.
Maharaj, A. (2013). An APOS analysis of natural science students’ understanding of derivatives, South African Journal of Education, 33(1), 146-164.
Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view, Educational studies in mathematics, 61, 11-38.
Palha, S., Dekker, R., Gravemeijer, K. & van Hout-Wolters, B. (2013). Developing shift problems to foster geometrical proof and understanding, Journal of mathematical behavior, 32, 142-159.
Park, J., Park, M.-S., Park, M., Cho, J. & Lee, K.-H. (2013). Mathematical modelling as a facilitator to conceptualization of the derivative and the integral in a spreadsheet environment, Teaching mathematics and its applications, 32, 123-139.
Pedemonte, B. & Reid, D. (2011). The role of abduction in proving processes, Educational studies in mathematics, 76, 281-303.
Peirce, C. S. (C.P.) (1931–1935, 1958) Collected Papers of Charles Sanders Peirce. Cambridge, MA:Harvard UP.
Peng, Y. & Reggia, J. A. (1990). Abductive inference models for diagnostic problem-solving. New York: Springer.
Prawat, R. S. (1999). Dewey, Peirce, and the learning paradox, American educational research journal, 36(1), 47-76.
Presmeg, N. (2005). Metaphor and metonymy in processes of semiosis in mathematics education. In M. H. G. Hoffinann, J. Lenhard & F. Seeger (Eds.) Activity and Sign - Grounding Mathematics Education (pp. 105-116). New York: Springer.
Rodríguez, O. H. & Fernández, J. M.(2010). A semiotic reflection on the didactics of the chain rule, The Montana mathematics enthusiast, 7(2&3), 321-332.
Sahin, Z., Aydogan-Yenmez. A. & Erbas, A. K. (2015). Relational understanding of the derivative concept through mathematical modeling: A case study. Eurasia Journal of Mathematics, Science & Technology Education, 11(1), 177-188.
Sfard, A. (2008). Thinking as communicating, human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.
Stewart, J. (2012). Calculus: Early transcendentals. Pacific Grove, CA: Brooks/Cole.
Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. London: Sage publications.
Tall, D. O. (1992). Students’ Difficulties in Calculus. Plenary Address. In Proceedings of Working Group 3 on Students‟ Difficulties in Calculus, ICME-7 (pp. 13-28), Québec.
Tall, D., Smith, D. & Piez, C. (2008) Technology and calculus. In M. Kathleen Heid & Glendon M. Blume (Eds.) Research on Technology and the Teaching and Learning of Mathematics, vol 1: Research Syntheses (pp. 207–258).
Uygur, T. & Ozdas, A. (2007). The effect of arrow diagrams on achievement in applying the chain rule. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 17(2), 131-147.
Watson, A. & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples, Mahwah, NJ: Lawrence Erlbaum Associates.

How to cite this article

APA

Park, J., & Lee, K.-H. (2016). How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling. Eurasia Journal of Mathematics, Science and Technology Education, 12(9), 2331-2352. https://doi.org/10.12973/eurasia.2016.1289a

Vancouver

Park J, Lee KH. How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling. EURASIA J Math Sci Tech Ed. 2016;12(9):2331-52. https://doi.org/10.12973/eurasia.2016.1289a

AMA

Park J, Lee KH. How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling. EURASIA J Math Sci Tech Ed. 2016;12(9), 2331-2352. https://doi.org/10.12973/eurasia.2016.1289a

Chicago

Park, JinHyeong, and Kyeong-Hwa Lee. "How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling". Eurasia Journal of Mathematics, Science and Technology Education 2016 12 no. 9 (2016): 2331-2352. https://doi.org/10.12973/eurasia.2016.1289a

Harvard

Park, J., and Lee, K.-H. (2016). How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling. Eurasia Journal of Mathematics, Science and Technology Education, 12(9), pp. 2331-2352. https://doi.org/10.12973/eurasia.2016.1289a

MLA

Park, JinHyeong et al. "How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling". Eurasia Journal of Mathematics, Science and Technology Education, vol. 12, no. 9, 2016, pp. 2331-2352. https://doi.org/10.12973/eurasia.2016.1289a